Rotation suppresses giant-scale solar convection

Abstract

The observational absence of giant convection cells near the Sun's outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller dominant length scales. We specifically predict a characteristic convection length scale of roughly 30-Mm throughout much of the convection zone, implying weak flow amplitudes at 100- to 200-Mm giant cells scales, representative of the total envelope depth. Our reasoning is such that Coriolis forces primarily balance pressure gradients (geostrophy). Background vortex stretching balances baroclinic torques. Both together balance nonlinear advection. Turbulent fluxes convey the excess part of the solar luminosity that radiative diffusion cannot. We show that these four relations determine estimates for the dominant length scales and dynamical amplitudes strictly in terms of known physical quantities. We predict that the dynamical Rossby number for convection is less than unity below the near-surface shear layer, indicating rotational constraint.

Keywords: Rossby number; differential rotation; rapid rotation; solar convection.

Fig. 1.

(A) The estimated dynamical Rossby number as a function of depth. (B) The estimated convective flow speed estimates as a function of depth. In both panels, the solid curves represent the rapidly rotating regime (CASE II). The dashed lines show the slowly rotating counterfactual (CASE I). Both ${\text{Ro}}_{\text{d.}\hspace{0.17em}}$ estimates are less than unity for much of the convection zone; hence, rotating assumptions apply. The shaded regions above $r/R_{\odot }\approx 0.93$ mark where rotational effects are subdominant. Several other effects become important in the outer layers.

Fig. 2.

The convective length-scale estimate as a function of depth in the solar convection zone. The solid red curve shows the estimate under the rapidly rotating assumption (CASE II). The dashed line shows the ${\mathit{\text{H}}}_{\rho }\left(r\right)$ profile from Eq. 11 (CASE I). The shaded region above $r/R_{\odot }\approx 0.93$ marks where rotation is subdominant (i.e., a depth $\approx 50\text{Mm}$). The rotationally constrained length scale stays consistently $\approx 30-\text{Mm}$ and is the length scale realized below $r/R_{\odot }\approx 0.93$. The density-scale height becomes the dominant length scale only in the near-surface regions.

Fig. 3.

The local Rossby number computed from helioseismic-inferred profile $\mathrm{\Omega }\left(r,\theta \right)$ in the convection zone. We use a fourth-order finite difference to compute the gradient from the raw rotation rate data, which are found in the electronic supplementary material from Larson and Schou (92).

Fig. 4.

The local thermal-wind entropy profile computed from helioseismic-inferred profile $\mathrm{\Omega }\left(r,\theta \right)$ in the convection zone. We use a fourth-order finite difference to compute ${\partial }_z=\cos \theta {\partial }_r-r^{-1}\sin \theta {\partial }_{\theta }$ and a trapezoidal rule to integrate over $\theta$. The data are found in the electronic supplementary material from Larson and Schou (92).

Fig. 5.

Radial entropy profiles. The orange curve is the $\theta$ rms of the two-dimensional profile from Fig. 4. The black curve is the convective estimate regime based on the flux debt; ${\left(2\mathrm{\Omega }{\mathit{\text{H}}}_{\rho }{F}_0^3\right)}^{1/5}/h_0$ with background enthalpy $h_0=c_p{T}_0$. The dashed curve shows the estimate based on nonrotating theory, $F_0^{2/3}/h_0$. By magnitude, all three of these profiles are consistent. Distinguishing rotating vs. nonrotating theories lies with other quantities, particularly the convective length scale.